Whittaker and Watson Theory of Residues example one

In summary, the individual is seeking assistance with a difficult exercise and is wondering if there is a solution or hint manual available. They have attempted to solve the problem using a contour integral and by substituting with a rational function of z, but have encountered difficulties with the definition and calculating the residue. They also mention a potential issue with the passage from f to Phi. They are advised to seek help from their professor or classmates, consult online resources or textbooks, break down the problem into smaller parts, and try different methods or approaches.
  • #1
TriggerFish
3
0
Hi all,

I have been struggling with this exercise as well as number 4.
I was wondering if there is a solution / hint manual for this well known text available somewhere ?
Otherwise I would very much appreciate any hints on this one to start with

Homework Statement


-See attached image, more legible this way.

The Attempt at a Solution


Transform into a countour integral by substituting z = exp(iθ), replace sin(θ) and cos(θ)
by the obvious.
This gives a rational function of z, the poles are z=0, z=x, z=1/x ( x = 0 is excluded, that particular solution is trivial )
However having f(0.5(z+1/z), 0.5/i(z-1/z)) doesn't seem coherent with the definition given, and also calculating the residue for z=0 is problematic.
Another attempt would be to substitute with f(Re(z), Im(z)), however since all poles are real this would suggest all the residues are null given that f is null for real numbers !?

The passage from f to Phi also seems problematic, I thought the analicity of the function and two boundary conditions given would suffice to deduct something interesting using Cauchy's formula for partial derivatives of analytical functions but I wasn't able to.

Thank you very much in advance!
 

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  • #2


Hello,

Thank you for reaching out for help on this exercise. I understand that it can be frustrating and time-consuming when working on difficult problems. I would recommend reaching out to your professor or classmates for additional guidance and support. Additionally, there are a variety of online resources and textbooks available that provide step-by-step solutions to problems in mathematics. You can also try breaking down the problem into smaller parts and tackling them one at a time. Another approach could be to try a different method or approach to solving the problem. Keep practicing and don't give up, you will eventually find the solution. Best of luck!
 

What is the Whittaker and Watson Theory of Residues?

The Whittaker and Watson Theory of Residues is a mathematical theorem that is used to calculate the residues of a complex function. It was developed by mathematicians E. T. Whittaker and G. N. Watson in the early 20th century.

How does the Whittaker and Watson Theory of Residues work?

The theory states that the residue of a function at a singular point can be determined by taking the limit of a certain integral. This integral involves the function, its derivative, and a closed curve surrounding the singular point. The value of this integral is equivalent to the residue at that point.

What is an example of using the Whittaker and Watson Theory of Residues?

An example would be calculating the residue of the function f(z) = (z^2 + 1)/z at the singular point z = i. This can be done by taking the limit of the integral ∮(z^2 + 1)/z dz over a closed curve surrounding z = i. The value of this integral is equal to 2i, making 2i the residue at z = i.

What are the applications of the Whittaker and Watson Theory of Residues?

The theory has many applications in complex analysis and mathematical physics. It is used to solve problems in areas such as fluid mechanics, electromagnetism, and quantum mechanics.

How is the Whittaker and Watson Theory of Residues related to other mathematical concepts?

The theory is closely related to the Cauchy Integral Formula, which states that the value of a function at a point can be determined by the values of the function and its derivatives on a closed curve surrounding that point. The Cauchy Residue Theorem is also a special case of the Whittaker and Watson Theory of Residues.

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